TY - JOUR

T1 - On non-convex quadratic programming with box constraints

AU - Burer, S

AU - Letchford, A N

PY - 2009

Y1 - 2009

N2 - Nonconvex quadratic programming with box constraints is a fundamental NP-hard global optimization problem. Recently, some authors have studied a certain family of convex sets associated with this problem. We prove several fundamental results concerned with these convex sets: we determine their dimension, characterize their extreme points and vertices, show their invariance under certain affine transformations, and show that various linear inequalities induce facets. We also show that the sets are closely related to the Boolean quadric polytope, a fundamental polytope in the field of polyhedral combinatorics. Finally, we give a classification of valid inequalities and show that this yields a finite recursive procedure to check the validity of any proposed inequality.

AB - Nonconvex quadratic programming with box constraints is a fundamental NP-hard global optimization problem. Recently, some authors have studied a certain family of convex sets associated with this problem. We prove several fundamental results concerned with these convex sets: we determine their dimension, characterize their extreme points and vertices, show their invariance under certain affine transformations, and show that various linear inequalities induce facets. We also show that the sets are closely related to the Boolean quadric polytope, a fundamental polytope in the field of polyhedral combinatorics. Finally, we give a classification of valid inequalities and show that this yields a finite recursive procedure to check the validity of any proposed inequality.

KW - non-convex quadratic programming

KW - global optimisation

KW - polyhedral combinatorics

KW - convex analysis

U2 - 10.1137/080729529

DO - 10.1137/080729529

M3 - Journal article

VL - 20

SP - 1073

EP - 1089

JO - SIAM Journal on Optimization

JF - SIAM Journal on Optimization

SN - 1095-7189

IS - 2

ER -